Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\).
Let \(p_w(n)\) be the cardinality of \(P_w(n)\).
Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).
\(|P_{f(1IQM_1)}(2) \setminus P_{f(6UGC_1)}(2)|=171\),
\(|P_{f(6UGC_1)}(2) \setminus P_{f(1IQM_1)}(2)|=3\).
Let
\(
Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)|
\)
be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:1111000001001101111000001101101100101101100100100101011000000001101100101110000100000010111101001101010111101100011000110000111011100000100000101101101000000100011100011011000000010010011101001000011011101101010010011000101110110001000
Pair
\(Z_2\)
Length of longest common subsequence
1IQM_1,6UGC_1
174
1
1IQM_1,3LZW_1
156
4
6UGC_1,3LZW_1
196
2
Newick tree
[
6UGC_1:97.06,
[
1IQM_1:78,3LZW_1:78
]:19.06
]
Let d be the
Otu--Sayood
distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{244
}{\log_{20}
244}-\frac{9}{\log_{20}9})=82.0\)
Status
Protein1
Protein2
d
d1/2
Query variables
1IQM_1
6UGC_1
107
54.5
Was not able to put for d Was not able to put for d1
In notation analogous to
[Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]